Fuzzy cardinality based evaluation of quantified sentences

Fuzzy cardinality based evaluation ofquanti®ed sentences

Miguel Delgado *, Daniel S�anchez, Maria Amparo Vila

Department of Computer Science and Arti®cial Intelligence, E.T.S. de Ingenier�õa Inform�atica,

University of Granada, Avda. de Andalucia 38, 18071 Granada, Spain

Received January 1999; accepted June 1999

Abstract

Quanti®ed statements are used in the resolution of a great variety of problems.

Several methods have been proposed to evaluate statements of types I and II. The

objective of this paper is to study these methods, by comparing and generalizing them.

In order to do so, we propose a set of properties that must be ful®lled by any method of

evaluation of quanti®ed statements, we discuss some existing methods from this point of

view and we describe a general approach for the evaluation of quanti®ed statements

based on the fuzzy cardinality and fuzzy relative cardinality of fuzzy sets. In addition,

we discuss some concrete methods derived from the mentioned approach. These new

methods ful®ll all the properties proposed and, in some cases, they provide an inter-

pretation or generalization of existing methods. Ó 2000 Elsevier Science Inc. All rights

reserved.

1. Introduction

Quanti®ed sentences are used in a large number of applications for repre-senting assertions and/or restrictions about the number or percentage of ob-jects that verify a certain property. These assertions and/or restrictions are oneof the most used by humans in their reasoning processes. Because of this, some

International Journal of Approximate Reasoning 23 (2000) 23±66

*Corresponding author. Tel.: +34-958-243194; fax: +34-958-243317.

E-mail addresses: [email protected] (M. Delgado), [email protected] (D. SaÂnchez),

[email protected] (M.A. Vila)

0888-613X/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved.

PII: S 0 8 8 8 - 6 1 3 X ( 9 9 ) 0 0 0 3 1 - 6

authors have tried to de®ne a mathematical model for the representation of thisknowledge in the ®eld of AI by using the theory of fuzzy sets. The ®rst ap-proach was described in [28] by Zadeh. Since then, quanti®ed sentences havebeen used in the resolution of several problems. One of the ®elds wherequanti®ed sentences have been more applied is that of ¯exible database que-rying. There is a large amount of literature on this topic, such as [3,7,15,17].Quanti®ed sentences have been applied in other ®elds such as pattern recogi-tion, inductive learning, aggregation and decision making among others. Pa-pers as [28±30,18,8±11,25] are some examples. Applications of quanti®edsentences in the ®eld of expert systems are discussed in [13], where there is asection devoted to the applications of quanti®ed sentences. In the ®eld of datamining, quanti®ed sentences have been used for example in [22]. We will usequanti®ed sentences to develop data mining applications. In general, quanti®edsentences are a useful approximate reasoning tool for solving problems wherelinguistic quanti®ers and natural languages are used in the representation ofour knowledge.

Quanti®ed sentences are usually classi®ed into two classes, called type Isentences and type II sentences. A type I sentence is a sentence of the form:

Q of X are A;

where X � fx1; . . . ; xng is a ®nite set, Q a linguistic quanti®er and A a fuzzyproperty de®ned over X.

A type II sentence can be described in general as:

Q of D are A;

where D is also a fuzzy property over X. Obviously, type I sentences are aspecial case of type II sentences where D � X . The following are examples ofeach type of sentences:

Type I : Most of the students are young:

Type II : Most of the efficient students are young:

In these examples, the set X is a ®nite set of students, the quanti®er is ``Most'',the set A is the property ``young'' and the set D is the property `è�cient''.

Two kinds of linguistic quanti®ers are taken into consideration in theevaluation of quanti®ed sentences: these are called absolute and relativequanti®ers. They are de®ned as possibility distributions, over the non-negativeintegers and the real interval �0; 1�, respectively. Absolute quanti®ers representfuzzy integer quantities or intervals. Examples of this type are `àpproximately5'' and ``between 2 and 4''. Relative quanti®ers represent fuzzy proportions.Some examples are ``Many'', ``Most'' and `Àll''. Although relative quanti®ersare de®ned over the real interval �0; 1� for simplicity, in fact only values of therational interval �0; 1� are used in the evaluation.

24 M. Delgado et al. / Internat. J. Approx. Reason. 23 (2000) 23±66

There are four possible combinations (type of sentence, type of quanti®er).Each possible combination is evaluated in a slightly di�erent way by any ex-isting method of evaluation. In [12] sentences and quanti®ers are related asfollows: (type I, absolute); (type II, relative). In fact, one type II sentence withan absolute quanti®er can be transformed into an equivalent type I sentence inthe following way: ``Q of D are A'' with Q an absolute quanti®er is equivalentto ``Q of X are A \ D''. But in our opinion, it is important to add the pair (typeI, relative) to the problem of the evaluation of sentences. Some of the betterstudied methods of evaluation focus on this case.

Evaluation of quanti®ed sentences tries to obtain an accomplishment degreein the real interval �0; 1� for the sentence. Di�erent methods have been pro-posed to perform the evaluation of quanti®ed sentences following this ap-proach. Methods for the evaluation of type I sentences are described in[30,18,19,1,2,4]. Methods for type II sentences are described in [30,21,17,4,5].We will talk brie¯y about these methods in this paper. Some other methodsobtain a real interval or a fuzzy set as the accomplishment degree for thesentences, see [14] for example, and will not be mentioned in this paper. Our®rst objective in this work is to de®ne what we consider some appropriateproperties for any method of evaluation of quanti®ed sentences that obtains areal value as the accomplishment degree, and to study and compare the existingmethods from this point of view. The ®nal objective is to de®ne new methods toperform the evaluation according to the properties de®ned.

The contents of the paper are structured as follows. In Section 2, we give aset of properties to be ful®lled by any method of evaluation. In Section 3, weshow the existing methods for the evaluation of type I and type II sentences.Section 4 is devoted to the description of the approach we use to de®ne newmethods, along with previous de®nitions of cardinalities of fuzzy sets and theirproperties. Section 5 shows new methods for the evaluation of type I sentences.In Section 6, we de®ne new methods for the evaluation of type II sentences.Section 7 contains our conclusions and future work.

2. Appropriate properties for sentence evaluation methods

Every existing method of evaluation is de®ned according to a di�erentapproach or measure. The validity of any method comes from the seman-tic validity of the selected approach or measure when performing theevaluation. Despite this, any method is required to ``work well'' in the sensethat the results obtained are somehow appropriate and coherent with what weexpect. In this section, we propose what we consider some appropriateproperties to be ful®lled by any method of evaluation. They are not intended tobe a closed set of properties but a collection of known cases and intuitiveconstraints.

M. Delgado et al. / Internat. J. Approx. Reason. 23 (2000) 23±66 25

2.1. Properties for the evaluation of type I sentences (Q of X are A)

Property 2.1.1 (Crisp case). If A is crisp, then the (known) result of the evalu-ation must be

QAj jXj j

� �if Q is relative, and

Q Aj j� �if Q is an absolute quantifier.

Property 2.1.2. Evaluation must be coherent with fuzzy logic in the case ofquantifiers ``exist'' and ``all''. The sentence ``Q of X are A'' with Q� $ can berepresented and evaluated using fuzzy logic asW

xi 2 XA�xi�

with the fuzzy union performed by a t-conorm (usually the maximum), and in thecase Q � 8 the evaluation must beV

xi 2 XA�xi�

with the fuzzy intersection performed by a t-norm.

Property 2.1.3. Evaluation must be coherent with quantifiers inclusion. GivenQ � Q0 (Q is more restrictive than Q0), Eval (``Q of X are A'') 6 Eval (``Q0 of X areA''), where Eval (c) with c a quantified sentence is the result obtained from theevaluation of c. Intuitively, it is more difficult to fit Q than to fit Q0 in the evaluation.

Property 2.1.4. Evaluation must be time-efficient (as much as possible). Weconsider time-efficient an efficiency between O�n� and O�n logn�, n � jX j.

Property 2.1.5. Evaluation must not be too ``strict'', i.e. given a quantifier definedover the set H � fp=qjp 2 f0; . . . ; ng; q 2 f1; . . . ; ngg, with Q 6� ; and Q 6� H ,we must be able to find a fuzzy set A so that the evaluation of the sentence is notin f0; 1g. The convenience of this property and the problems that can be derived ifwe do not require, it can be seen in Section 3.1.

Property 2.1.6. Evaluation must allow us to use any quantifier, i.e. any possibilitydistribution over the non-negative integers or over the real interval �0; 1�. Thereare many quantifiers with clear semantics that fall outside the group of ``coherentquantifiers'' used by many methods (see Section 3.2). We shall see an example inthis work.


2.2. Properties for the evaluation of type II sentences (Q of D are A)

Property 2.2.1 (Crisp case). If A and D are crisp, then the (known) result of theevaluation must be

QA \ Dj j

Dj j� �

:

We assume Q is relative.

Property 2.2.2. In the case D � X and for relative quantifiers, the resultingevaluation method is a valid method for the evaluation of type I sentences. Type Isentences are, in fact, a special case of type II sentences where D � X , so in thiscase the evaluation method of type II sentences must be a valid evaluation methodof type I sentences.

Property 2.2.3. Evaluation must be time-efficient (as much as possible), i.e. O�n�or O�n logn�.

Property 2.2.4. If D � A and D is a normal set then the evaluation method mustreturn the value Q (1). This is an intuitive property (the percentage of D that areA is 100%).

Property 2.2.5. If D \ A � ; then the evaluation method must return thevalue Q (0). This is also an intuitive property (the percentage of D that are A is0%).

Property 2.2.6. Evaluation must be coherent with fuzzy logic in the case of thequantifier ``exist'' and ``all'', givingW

xi 2 XA�xi� ^ D�xi��

and Vxi 2 X

D�xi�� ! A�xi��

using some t-conorm for the union and a t-norm for the intersection, and! beinga fuzzy implication.

Property 2.2.7. Evaluation must allow us to use any quantifier, i.e. any possibilitydistribution over �0; 1�.


Property 2.2.8. Evaluation must not be too ``strict'', i.e. given a quantifierQ in the rational interval �0; 1� with Q 6� ; and Q 6� H � fp=q withp 2 f0; . . . ; ng and q 2 f1; . . . ; ngg we must be able to find fuzzy sets A and D sothat the evaluation of the sentence is not in f0; 1g.

3. Some existing methods for the evaluation of quanti®ed sentences

Given a quanti®ed sentence of type I ``Q of X are A'', with X � fx1; . . . ; xng a®nite crisp set and A a fuzzy set over X, the following are some existingmethods to perform the evaluation of the sentence.

3.1. Type I sentences

3.1.1. Zadeh's methodZadehÕs method [30] is based on the use of the non-fuzzy cardinality

R-count, also called power. In the case of relative quanti®ers, the ®nal evalua-tion is

ZQ�A� � QP �A�

Xj j� �

;

where P�A� is the power of A de®ned in Section 4.1.1. For absolute quanti®erswe have

ZQ�A� � Q P �A�k k� �;where jP�A�j is the integer part of the real number P �A�.

This method ful®lls all properties of Section 2.1 except Properties 2.1.5 and2.1.2. As a counterexample for Property 2.1.5, in the case of universallyquanti®ed sentences, the evaluation is 1 if and only if A � X , and 0 in any othercase. The case of the quanti®er $ is similar, the evaluation being 0 if and only ifA � ;. As a counterexample for Property 2.1.2 we have the caseA � f0=x1; 0:5=x2g for the quanti®er exists. The result obtained using ZadehÕsmethod is 1, but there is no t-conorm such that 0 _ 0:5 � 1 (every t-conormveri®es that 0 _ a � a). For the quanti®er all, one counterexample isA � f1=x1; 0:9=x2g. The result obtained by ZadehÕs method is 0, but there is not-norm such that 1 ^ 0:9 � 0 (every t-norm veri®es 1 ^ a � a). The e�ciency ofthe method is O�n�, n being equal to jX j.

3.1.2. Yager's method based on OWA operatorsThis method de®ned in [19] only considers the case of relative and non-

decreasing quanti®ers verifying Q�0� � 0, Q�1� � 1 (the so-called ``coherentquanti®ers''). A coherent family of quanti®ers is a set of quanti®ersfQ1; . . . ;Qng verifying Q1 � 8, Qn � 9 and Qi � Qi�1. By Property 2.1.2,


sentences with quanti®ers Q1 � 8 and Qn � 9 are evaluated by means of a t-norm and a t-conorm, usually min and max. The evaluation of sentences withother quanti®ers of a coherent family can be performed by means of an OWAoperator. Any OWA operator gives a result between min and max, and thecoe�cients of the operator are obtained from the quanti®er and the valuen � jX j in the following way, that guarantees Property 2.1.3:

wi � Q�i=n� ÿ Q��iÿ 1�=n�; i 2 f1; . . . ; ng and Q�0� � 0:

Finally, the evaluation of the sentence is

YQ�A� �Xn

i�1

wibi;

where bi is the ith largest value of belongingness to the fuzzy set A. In thefollowing, this will be the meaning of bi.

This method ful®lls every property of Section 2.1 except Property 2.1.6.Property 2.1.4 is powered if we consider that for every quanti®er Q of a co-herent family and every value n we calculate and save the values of the coef-®cients wi. If the values of A are arranged in descending order, the e�ciency isO�n�. If not, the best e�ciency is O�n logn�. Although Property 2.1.6 is notveri®ed by this method by the requirement of the quanti®er to be coherent,sentences with some kind (not all) of non-coherent quanti®ers can be evaluatedby means of semantic equivalences with sentences where the quanti®er is the``antonym'' of the original quanti®er and the fuzzy set is the complement of A.This method is described in [23,13].

3.1.3. Yager's non-OWA family of methodsYager [18] proposes to perform the evaluation of type I sentences in the

following way:

Y 0Q�A� � maxC�X^1 Q

Cj jn

� �; ^2

xi2CA xi� �

� �:

The last expression is applied in the case of relative quanti®ers, and

Y 0Q�A� � maxC�X^1 Q Cj j� �; ^2

xi2CA xi� �

� �for absolute quanti®ers. In both cases, ^1 and ^2 are two t-norms.

Some concrete members of this family of methods are studied by Yager [18].Among them, we can remark the case where ^1 � ^2 � min. In this case, themethod obtained for relative quanti®ers is

Ym0Q�A� � maxC�X

min QCj jn

� �; min

xi2CA xi� �

� �:


The expression for absolute quanti®ers is similar. The properties of this methodwill be discussed in the next section.

3.1.4. Methods based on the Choquet and the Sugeno integralsThe use of the Choquet and the Sugeno integrals for the evaluation of

quanti®ed sentences is described among others in [1,2]. As in previous methods,the evaluation is restricted to the case of coherent quanti®ers. In the case ofrelative quanti®ers, the method based on ChoquetÕs integral is de®ned by theexpression

CQ�A� �Xn

i�1

bi � �Q�i=n� ÿ Q��iÿ 1�=n�

and the method based on SugenoÕs integral is expressed as

SQ�A� � max16 i6 n

min�Q�i=n�; bi�;

where, as in previous cases, bi is the ith largest value of A.The following properties hold:

Property 3.1.4.1. The method based on the Choquet integral is the OWA-basedmethod of Yager. This is obvious and is shown in [1,2].

Property 3.1.4.2. The method based on the Sugeno integral is the methodYm0Q of Section 3.3, as shown in [1], and hence is a member of the family ofmethods Y0Q.

Proof (Relative quantifier, for absolute quantifier is similar).

Ym0Q�A� � maxC�X

min QCj jn

� �; min

xi2CA xi� �

� �� max

16 i6 nmax

Cj j�imin Q

Cj jn

� �; min

xi2CA xi� �

� �� max

16 i6 nmin Q

in

� �; max

Cj j�iminxi2C

A xi� ��

� max16 i6 n

min Qin

� �; L A; i� �

� �� max

16 i6 nmin Q

in

� �; bi

� �� SQ�A�;

where L�A; i� is de®ned in [6] as the possibility that ``the cardinality of A is atleast i''. In the same paper, we show that L�A; i� � bi. We give the de®nition ofL�A; i� in Section 4.1.3, Property 4.1.3.1.


This method ful®lls Properties 2.1.2±2.1.5. The e�ciency is O�n logn� if A isnot arranged in decreasing order, and O�n� otherwise. Properties 2.1.1 and2.1.6 are in con¯ict, because Property 2.1.1 is ful®lled only if the quanti®er iscoherent. The following is a counterexample: let X � fx1; x2; x3g andA � f1=x1; 1=x2; 0=x3g and let Q�0� � 0; Q�1=3� � 1; Q�2=3� � 0; Q�1� � 0.Clearly, Q is not non-decrecient, and hence Q is not coherent. We then haveSQ�A� � maxfmin�1; 1�;min�0; 1�;min�0; 0�g � 1, while jAj � 2 and then theexpected result by Property 2.1.1 must have been Q�2=3� � 0, so Property 2.1.1is not ful®lled when Q is not coherent.

3.2. Type II sentences

The evaluation of type II sentences is slightly more complex than the eval-uation of type I ones. There are fewer methods for type II sentences than fortype I sentences. Given a quanti®ed sentence of type II Q of D are A, with Dand A two fuzzy sets over X, X being a ®nite set, the following are somemethods to perform the evaluation of the sentence.

3.2.1. Zadeh's methodZadehÕs method is described in [30], and can be seen as an application of the

cardinality approach. The case D � ; is not evaluable. This method obtains therelative cardinality of A with respect to D as (see Section 4.2.1)

P �A=D� � P �A \ D�P �D� ;

where P is the power (R-count). The intersection is usually obtained via theminimum.

The evaluation of the sentence is, ®nally

ZQ�A=D� � Q P �A=D�� QP �A \ D�

P �D��

:

It is easy to prove that ZadehÕs method veri®es Property 2.2.1. When D � X ,we have ZQ�A=X � � ZQ�A�, so Property 2.2.2 is ful®lled. The e�ciency of themethod is O�n�, so Property 2.2.3 is also ful®lled. Properties 2.2.4, 2.2.5 and2.2.7 are easy to prove. Property 2.2.6 is not ful®lled, because we have shownthat in the case D � X , the remaining method for the evaluation of type Isentences is ZQ(A), and this method does not ful®ll the coherency with logic.The same counterexamples used in Section 3.1.1 are valid here. For the samereason, the method does not ful®ll Property 2.2.8. A counterexample similar tothat of ZadehÕs method for the evaluation of type I sentences can be shownusing the quanti®ers ``exists'' and ``all'' in the case D � X .


3.2.2. Yager's method based on OWAYagerÕs proposal is described in [21] and is based on the OWA operator

where the parameters wi are calculated from Q and D. This method is de®nedfor coherent and relative quanti®ers. The parameters of the OWA operator arecalculated as

wi � Q�Si� ÿ Q�Siÿ1� i 2 f1; . . . ; ng;where

Si � 1

d

Xi

j�1

ei and d �Xn

k�1

ek

and ek is the ith smallest value of belongingness to D and S0 � 0.The ®nal evaluation of the sentence is

YQ�A=D� �Xn

i�1

wici;

where ci is the ith largest value of belongingness to the fuzzy set qD _ A.The method does not ful®ll Property 2.2.5. As a counterexample,

let A � f1=x1; 0=x2g and D � f0=x1; 0:7=x2g. Let Q � 9. The obtained resultusing YagerÕs method is 0.3, while the expected value was 0. Property 2.2.4 isnot ful®lled. As a counterexample, let A � f1=x1; 0:9=x2g and D �f1=x1; 0:5=x2g and Q�x� � x. Clearly D � A and D is normalized so the expectedvalue is Q�1� � 1, but the result obtained by YQ is 0.93. Property 2.2.6 is notful®lled by the method for the quanti®er `èxists'', and the last counterexampleis also a counterexample for this property (any t-norm t veri®es t�x; 0� �t�0; x� � 0 and there is no t-conorm tc such that tc�0; 0� � 0:3). Property 2.2.6for the quanti®er `àll'' is ful®lled by this method, using the minimum and theimplication qD _ A. Obviously, Property 2.2.7 is not ful®lled by this method.YagerÕs method ful®lls the rest of properties of Section 2.2, although the e�-ciency must be improved by storing values of wi for every tuple �Q;D; n�.

3.2.3. Method of Vila, Cubero, Medina and PonsThe main advantage of this method, described in [17], is the e�ciency O�n�,

together with a non-strict evaluation. The method uses the degree of `òrness''de®ned by Yager [19] for coherent quanti®ers. This value is de®ned in the realinterval �0; 1� and provides the degree of neighborhood of one quanti®er to thequanti®er $. By de®nition, orness�9� � 1 and orness�8� � 0. Any coherentquanti®er between $ and " has associated a degree in �0; 1�. Using the ornessand the logic evaluation of the sentences ``" of D are A'' and ``$ of D are A'',the evaluation of ``Q of D are A'' is given by

VQ�A=D� � oQ maxx2X�D�x� ^ A�x�� 1ÿ oQ�min

x2X�A�x� _ �1ÿ D�x��;


where oQ is the orness Q de®ned by

oQ �Xn

i�1

nÿ inÿ 1

� �� Q i=n� �� ÿ Q �i� ÿ 1�=n��:

This method only ful®lls Properties 2.2.3, 2.2.6 (easy to check), and 2.2.7.

4. The cardinality approach for the evaluation of sentences

Our approach for the evaluation of type I sentences is to obtain the accom-plishment degree of a sentence by means of the degree of compatibility betweenthe quanti®er and the cardinality of the fuzzy set A. The mentioned approach forthe evaluation of quanti®ed sentences is used for the evaluation of type I sen-tences. Type II sentences can be evaluated by obtaining the compatibility be-tween the ``relative cardinality'' of A with respect to D, and the relative quanti®erQ. The crisp relative cardinality of A with respect to D is the percentage of ele-ments of D that are elements of A. Some of the methods described in Section 3can be interpreted in terms of this approach as we shall see in Section 4.4.

In this approach, one method of evaluation of quanti®ed sentences is given bythree elements: the schema of representation of the cardinality of a fuzzy set, themethod of calculus of the cardinality, and the method for obtaining the com-patibility between cardinality and quanti®er. One usual way of representing thecardinality of a fuzzy set is by means of a scalar value, either integer or real.Another way of representation of the cardinality of a fuzzy set is the so-called``fuzzy cardinality''. This consists of representing the cardinality as a fuzzy setover the non-negative integers. Several methods to calculate the cardinalityusing one or another of these schemas have been developed. In Section 4.1, wewill look brie¯y at some of the most important existing methods related tosentence evaluation, and we also describe several new recently proposedmethods. Methods for the representation and calculation of the relative cardi-nality of fuzzy sets are also described in Section 4.2. We will brie¯y talk aboutthe calculus of the compatibility between cardinality and quanti®er in Section 4.3.

4.1. Cardinality of a fuzzy set

The following are some measures of the cardinality of a fuzzy set.

4.1.1. Power (R-count)This is an example of a scalar-valued measure of the cardinality of a fuzzy

set. This measure was de®ned by De Luca and Termini. Given a fuzzy set Aover a ®nite set X � fx1; . . . ; xng, the Power of A, P �A�, is de®ned as

P �A� �Xxi2X

A�xi�:


4.1.2. Zadeh's ®rst methodThe fuzzy cardinality Z(A) is de®ned as follows:

Z�A; k� � 0 if does not exist a jAaj � k;jsup a jAaj � kjf g otherwise:

�

4.1.3. Method EDThis method is de®ned in [6] as a member of a more general family of

cardinalities.

De®nition 4.1.3.1. Let X � fx1; . . . ; xng and A a fuzzy set over X. First, wede®ne the possibility that at least k elements of X belong to A, L�A; k�, as

L�A; k� �1; k � 00; k > n�

�i1;...;ik�2Ik

�A�xi1� � � � A�xik ��; 16 k6 n;

8><>:where Ik is the set of k-tuples of indexes de®ned by

Ik � �i1; . . . ; ik� j i1

�< i2 < � � � < ik with ij 2 f1; . . . ; ng 8j 2 f1; . . . ; kg

and Å and are a t-conorm and a t-norm, respectively.

Property 4.1.3.1. Let Å and be the maximum and the minimum, respectively.Then

L�A; k� � bk 8k 2 f1; . . . ; ng;where bk is the kth largest value of belongingness of an element to the fuzzy set A.

Proof. Every t-norm is non-decreasing, so the largest value between the ex-pressions bA�xi1�; . . . ;A�xik �c will be bb1; . . . ; bkc � bk, because we are usingthe minimum as t-norm. We are using the maximum as t-conorm, soL�A; k� � bk.

De®nition 4.1.3.2. We de®ne the possibility that exactly k elements of X belongto A, E�A; k�, as

E�A; k� � L�A; k� L�A; k � 1�;where is any t-norm and the bar stands for a fuzzy complement.

The expression of E de®nes a family of fuzzy cardinalities. Some existingmethods, such as the Dubois±Prade method, ZadehÕs FECount and RalescuÕs


method, are proved to be members of the family E in [6]. The following is a newmethod of the family E de®ned in the same paper.

De®nition 4.1.3.3. The fuzzy cardinality ED is a member of the family E thatemploys the maximum and minimum in the de®nition of L and LukasiewiczÕs t-norm and the standard negation in the de®nition of E. The expression of themethod is

ED�A; k� � bk ÿ bk�1

with b0 � 1 and bn�1 � 0.

Proof. By Property 4.1.3.1 when using max±min with L, we have L�A; k� � bk.Using in E LukasiewiczÕs t-norm and the standard negation we have

ED�A; k� � maxfbk � �1ÿ bk�1� ÿ 1; 0g� maxfbk ÿ bk�1; 0g� bk ÿ bk�1: �

As pointed out in [6], this method can be interpreted as a probabilisticmeasure of the cardinality of A, while other methods such as Zadeh's ®rstmethod (see Section 4.1.2) are possibilistic measures.

Property 4.1.3.3. The method ED verifiesXn

i�0

ED�A; i� � 1:

Proof.Xn

i�0

ED�A; i� � �b0 ÿ b1� � �b1 ÿ b2� � �b2 ÿ b3� � � � � � �bnÿ2 ÿ bnÿ1�

� �bnÿ1 ÿ bn� � �bn ÿ bn�1�� b0 ÿ bn�1 � 1: �

4.1.4. The Dubois±Prade methodDubois and Prade de®ne the set of crisp representatives of a fuzzy set as

R�A� � S A1 � S � Support�A�jf g:This set is an alternative representation of a fuzzy set by a set of crisp setsdi�erent to that of the representation theorem based on a-cuts, but verifyingthe same properties. The degree of representativity of a given set S of R�A� is


pA�S� � inf lA�u� u 2 Sjf g S 2 R�A�0 S 62 R�A�:

�Finally, the fuzzy cardinality of the fuzzy set A is given by

DP�A; k� � sup pA�S� Sj jjf � kg k 2 1; . . . ; nf g:In [6], we show that one alternative de®nition of DP is as follows:

DP�A; k� � 0; k < A1j j;bk; k P A1j j:

�

4.2. Relative cardinality of fuzzy sets

The relative cardinality of one set A with respect to a set D is a measure ofthe percentage of elements of D that are also elements of A. In general, it can bedescribed as follows:

Rel Card �A=D� � Card �A \ D�Card �D� :

4.2.1. Zadeh's methodZadeh de®nes the relative cardinality of a fuzzy set A with respect to a fuzzy

set D as:

P �A=D� � P �A \ D�P �D� ;

where P is the Power de®ned in Section 4.1.1, and the intersection is performedby means of the minimum.

4.2.2. Method ESThis method is de®ned in [6]. Let A and D be two fuzzy sets over X, D being

a normal fuzzy set. Let

M�A� � a 2 �0; 1� 9xi 2 X such that A�xi�jf � agand let

M�A=D� � M�A \ D� [M�D�and let

CR�A=D� � A \ D� �a��

Daj j such that a 2 M�A=D��

:


Then, the relative cardinality of A with respect to D, ES�A=D�, is de®ned as

ES�A=D; c� � max a 2 M�A=D� cj�

� A \ D� �a��

Daj j�8c 2 CR�A=D�:

If D is not a normal fuzzy set, we ®rst normalize D and scale the fuzzy set A\Dusing the same factor used in the normalization of D, before we begin theprocess.

4.2.3. Method ERThis method is also de®ned in [6]. Let A and D be two fuzzy sets over X, D

being a normal fuzzy set. Let M�A=D� � fa1; . . . ; amg be the set of represen-tative a-cuts de®ned in the last section, with 1 � a1 < a2 < � � � < am < am�1

� 0. Let

C�A=D; ai� �j A \ D� �ai

jjDai j:

Then, the relative cardinality of A with respect to D, ER�A=D�, is de®ned as

ER�A=D; c� �X

c�C�A=D;ai�ai� ÿ ai�1� 8c 2 CR�A=D�:

If D is not a normal fuzzy set, we ®rst normalize D and scale the fuzzy set A\Dusing the same factor used in the normalization of D, before we begin theprocess.

As pointed out in [6], this method can be interpreted as a probabilisticmeasure of the relative cardinality between A and D, while method ES is apossibilistic one.

4.3. Compatibility of fuzzy sets

The way we obtain the degree of compatibility between cardinality andquanti®er depends on the schema we are using to represent the cardinality.When we use a scalar value, the compatibility is obtained by evaluating thequanti®er at the point given by the cardinality. The way we perform this musttake into account the type of the scalar value (integer or real) and the type ofquanti®er (absolute or relative). Absolute quanti®ers can be evaluated for in-teger values, and relative quanti®ers can be evaluated for real ones. However,we can obtain the compatibility between an integer cardinality c and a relativequanti®er Q by evaluating Q�c=jX j�.

When we use a fuzzy set, the degree of compatibility between the cardinalityand the quanti®er is usually calculated as

�i2 0;...;nf g

�Q�i� C�A; i��


for type I sentences with absolute quanti®ers, where C�A; i� is the possibilitythat i is the cardinality of A, and and Å are a t-norm and a t-conorm, re-spectively. Similarly for type II sentences with relative quanti®ers, we have

�c��p=q� p;q2Z

�Q�c� C�A=D; c��;

where C�A=D; c� is the possibility that c is the relative cardinality of A withrespect to D, and

�i2 0;...;nf g

�Q�i=n� C�A; i��

for type I sentences with relative quanti®ers (we assume thatC�A=X ; i=n� � C�A; i�, with n � jX j).

4.4. Interpretation of some existing methods in terms of the cardinality approach

4.4.1. Zadeh's method for type I sentencesZadehÕs method represents the cardinality of A by means of a scalar value.

The cardinality is calculated by means of the power of A (Section 4.1.1). Fi-nally, the compatibility between cardinality and quanti®er is obtained evalu-ating the quanti®er at the point given by the cardinality.

4.4.2. Yager's method based on OWA for type I sentencesThis method (and hence the method based on the Choquet fuzzy integral)

can be interpreted in terms of the cardinality approach. The cardinality used isthe method ED of De®nition 4.1.3.3. We obtain the compatibility degree be-tween the quanti®er and the fuzzy cardinality ED by means of the Lukasiewiczt-conorm l�x; y� � minfx� y; 1g and the product as the t-norm. The interpr-etation is shown in Section 5.2, Property 5.2.1 of this paper, and the obtainedmethod is called GD. This method is equal to YagerÕs method only for co-herent quanti®ers, and is an extension that allows the use of any other quan-ti®er.

4.4.3. Method based on the Sugeno fuzzy integralThis method can be interpreted as the compatibility degree between the

fuzzy cardinality of Dubois±Prade (see Section 4.1.4) and the quanti®er, byusing max±min composition as follows


min�Q�i=n�;DP�A; k��:

We assume, as does the method based on the Sugeno fuzzy integral, that thequanti®er is coherent.


Proof. We shall discuss two cases:(a) Let jA1j � 0: Then, DP�A; k� � bk for all k 2 f1; . . . ; ng, so

max16 i6 n min�Q�i=n�; DP�A; i�� max16 i6 n min�Q�i=n�; bi� � SQ�A�:(b) Let jA1j � c > 0. Then DP�A; c� � bc � 1 and DP�A; i� � 0 for all i < c.

Then

max16 i6 n

min�Q�i=n�;DP�A; i��

� max max16 i<c

min�Q�i=n�;DP�A; i��; maxc6 i6 n

min�Q�i=n�;DP�A; i��

� maxc6 i6 n

min�Q�i=n�;DP�A; i��:

On the other hand,


min�Q�i=n�; bi�

� max max16 i<c

min�Q�i=n�; 1�; maxc6 i6 n

min�Q�i=n�; bi��

:

As Q is coherent, Q is non-decrecient and then

max16 i<c

min�Q�i=n�; 1� < min�Q�c=n�; bc� � min�Q�c=n�; 1�;so

SQ�A� � maxc6 i6 n

min�Q�i=n�; bi�� max

c6 i6 nmin�Q�i=n�;DP�A; i��

� max16 i6 n

min�Q�i=n�;DP�A; i��: �

4.4.4. Zadeh's method for type II sentencesZadehÕs method represents the relative cardinality of A with respect to D by

means of a scalar value. The relative cardinality is calculated by means of themethod described in Section 4.2.1. Finally, the compatibility between the relativecardinality and the quanti®er is obtained by evaluating the quanti®er at the pointgiven by the cardinality, as was the case of ZadehÕs method for type I sentences.

5. New methods for the evaluation of type I sentences

5.1. The family G of methods based on the cardinality approach

This family of methods is related to the family E of cardinalities described inDe®nition 4.1.3.2. The family G of methods of evaluation of type I sentences isgiven by

GQ�A� � �i2 0;...;nf g

E�A; i�� Q�i��


for absolute quanti®ers, and

GQ�A� � �i2 0;...;nf g

E�A; i�� Q�i=n��

for relative quanti®ers. This is a direct application of the cardinality approach.

Property 5.1.1. The family G of methods verifies Property 2.1.1 of evaluation oftype I sentences.

Proof (Absolute quantifiers, for relative quantifiers is similar). If A is crisp, thenby the property of any method of the family E (see [6])

E�A; i� � 0; i 6� Aj j;1; i � Aj j;

�and by the properties of every t-norm

E�A; i� Q�i� � 0; i 6� Aj j;Q�i�; i � Aj j:

�Finally, by the properties of every t-conorm

GQ�A� � Q Aj j� �: �

Property 5.1.2. The family G of methods verifies Property 2.1.6, i.e. it does notrequire any property from the quantifier to be used in the evaluation.

Property 5.1.3. The family G of methods verifies Property 2.1.3, because anyt-norm and any t-conorm are non-decreasing functions in their arguments.

5.2. The method GD

This method belongs to the family G, and is based on the fuzzy cardinalitymethod ED of the family E (see De®nition 4.1.3.3). Using the product as thet-norm and the Lukasiewicz t-conorm in the expression of the compatibility G,the method GD is de®ned by

GDQ�A� �Xn

i�0

ED�A; i� � Q�i�


GDQ�A� �Xn

i�0

ED�A; i� � Q�i=n�

for relative quanti®ers.


Proof. When using product and the Lukasiewicz t-conorm, we obtain the ex-pression

GDQ�A� � minXn

i�0

ED�A; i�(

� Q�i�; 1

):

By Property 4.1.3.3Pn

i�0 ED�A; i� � 1 , and obviously Q�i� 2 �0; 1� 8i andED�A; i� 2 �0; 1� 8i, so

Pni�0 ED�A; i� � Q�i�6 1, and hence GDQ�A� �Pn

i�0 ED�A; i� � Q�i�: �

The proof is similar for relative quanti®ers.

Property 5.2.1. If Q is a relative and a coherent quantifier, then

GDQ�A� � YQ�A� � CQ�A�;

i.e. the method GD is the method of Yager based on the OWA operator. As aconsequence, we can see Yager's method based on OWA as belonging to thefamily G in the case of relative and coherent quantifiers. This allows us to give aninterpretation of Yager's method in terms of the cardinality approach.

Proof.

GDQ�A� �Xn

i�0

ED�A; i� � Q�i=n�

�Xn

i�0

bi� ÿ bi�1� � Q�i=n�

� b0� ÿ b1� � Q�0� � b1� ÿ b2� � Q�1=n� � � � � � bnÿ1� ÿ bn�� Q��nÿ 1�=n� � bn� ÿ bn�1� � Q�1�

� fknown bn�1 � 0gb0 � Q�0� � b1 � Q�1=n�� ÿ Q�0�� b2

� Q�2=n�� ÿ Q�1=n�� bn � Q �n�� ÿ 1�=n� ÿ Q�1�� fknown Q is coherent and hence Q�0� � 0g

�Xn

i�1

bi � Q i=n� �� ÿ Q �i� ÿ 1�=n�� YQ�A�: �

Hence the method GD can be seen as a generalization of YagerÕs methodbased on OWA that can be used with any quanti®er (and hence ful®llingProperty 2.1.6). This method also ful®lls Properties 2.1.1, 2.1.3 and 2.1.6 be-cause GD is a method of the family G, and it is easy to see that Property 2.1.5 isalso ful®lled. The e�ciency is O�n logn� if we do not have the fuzzy set A ar-ranged in decreasing order, and O�n� otherwise. Therefore, method GD ful®llsall properties of Section 2.1.


5.3. The method ZS

This method is based on the cardinality approach. The fuzzy cardinalitymeasure used is ZadehÕs ®rst method

Z�A; k� � 0 if does not exist a jAaj � k;jsup a jAaj � kjf g otherwise:

�The compatibility between cardinality and quanti®er is calculated by means ofthe max±min composition using the cardinality approach in the followingway:

ZSQ�A� � maxk2 0;...;nf g

min Z�A; k�;Q�k��



min Z�A; k�;Q�k=n��

for relative quanti®ers.It is easy to see that method ZS ful®lls Properties 2.1.5 and 2.1.6. Method

ZS ful®lls every property of Section 2.1, as we show by the following prop-erties.

Property 5.3.1. The method ZS fulfills Property 2.1.1.

Proof. Let A be a crisp set. Then

Z�A; k� � 1; k � Aj j;0; k 6� Aj j

�and then for absolute quanti®ers we have

ZSQ�A� � Q Aj j� �

and for relative quanti®ers

ZSQ�A� � QAj jXj j

� �: �


Proof. The quanti®ers $ and " are de®ned as follows:

9�x� � 0; x � 0;1 otherwise;

�8�x� � 1; x � 1;

0 otherwise:

�


Then

�8� ZS8�A� � maxk2 0;...;nf g

min Z�A; k�; 8�k=n��

� Z�A; n�� sup a Aaj jjf � ng� min A�xi� xi 2 Xjf g:

�9� ZS9�A� � maxk2 0;...;nf g


� maxk2 1;...;nf g


� maxk2 1;...;nf g

Z�A; k�

� maxk2 1;...;nf g

sup a j Aaj j� � k�

� maxk2 1;...;nf g

max A�xi� j AA�xi�� ÿ � k

�� max A�xi� xi 2 Xjf g: �

Property 5.3.3. The method ZS fulfills Property 2.1.3 because max and min arenon-decreasing functions in their arguments.

Property 5.3.4. Let

M�A� � a 2 �0; 1� 9xij 2 X such that A�xi�f � ag [ 1f g:Then the method ZS has the following alternative expression:

ZSQ�A� � maxa2M�A�

min a ; Q Aaj j� ��

for absolute quantifiers, and

ZSQ�A� � maxa2M�A�

min a ; QAaj jXj j

� �� for relative quantifiers. This equivalence avoids obtaining Z(A) explicitly.

Proof. Let S(A) be the support of Z(A), i.e.

S�A� � k 2 0; . . . ; nf g 9a 2 M�A� such that jAajjf � kg:Then


min Z�A; k�;Q�k��

� maxk2S�A�

min max a jAajjf� � kg;


Q�k�� maxk2S�A�

maxjAaj�k

min a; Q�k��

� maxa2M�A�

min a; Q jAaj� �� : �

The proof for relative quanti®ers is similar. The e�ciency of the calculationof the equivalent expression is O�n logn� if A is not arranged and O�n� oth-erwise. Hence, method ZS ful®lls Property 2.1.4.

Property 5.3.5. If Q is a coherent quantifier, then the method ZS is the method Sbased on the Sugeno integral, i.e.

ZSQ�A� � SQ�A� � Ym0Q�A�:Hence, the method ZS can be seen as a generalization of the method S that can beused with any quantifier (and hence fulfilling Property 2.1.6) without any conflictwith Property 2.1.1.

Proof. We shall discuss two cases:1. Let bi 6� bj 8i; j 2 1; . . . ; nf g. Then it is easy to see that

Z�A; k� � bk8k 2 f1; . . . ; ng. Under these conditions,


min Z�A; k�;Q�k=n��

� maxk2 1;...;nf g

min Z�A; k�;Q�k=n�� fbecause Q is coherent and hence Q�0� � 0g

� maxk2 1;...;nf g

min bk;Q�k=n�� SQ�A�: �

2. Let h be the number of groups of repeated values of bi. Let li be the lengthof the group i, i 2 f1; . . . ; hg. Let ci be the ®rst index of the group i. Then thevalues of belongingness to A arranged in decreasing order are as follows:

fb1; . . . ; bc1; . . . ; bc1�l1

; . . . ; bc2; . . . ; bc2�l2

; . . . ; bch ; . . . ; bch�lh ; . . . ; bngwith bci � bci�1 � � � � � bci�liÿ1 � bci�li 8i 2 1; . . . ; hf g:

It is easy to see that

Z�A; k� �bk; k 2 BH � f1; . . . ; c1 � l1; c2 � l2; . . .

�only ci � li� . . . ; ch � lh; . . . ; ng;0 otherwise;

8><>:because in fact, Z�A; k� calculate the cardinality of every a-cut of A and assignsto that cardinality the possibility a, and all possible a-cuts of A are of the formAbj with j 2 BH.


Then, we have

SQ�A� � maxk2f1;::;ng

min�Q�k=n�; bk�� max

k2BHmin�Q�k=n�; bk�fbecause Q is coherent and hence Q is non-decrecientg

� maxk2BH

min�Q�k=n�; Z�A; k�� max

k2f1;...;ngmin�Q�k=n�; Z�A; k��

� fbecause if Z�A; k� � 0;min�Q�k=n�; Z�A; k�� 0

and the final maximum calculation does not changeg� max

k2f0;...;ngmin�Q�k=n�; Z�A; k��

� fbecause Q is coherent and henceQ�0� � 0;

and therefore the maximum does not changeg� ZSQ�A�: �

The following is a counterexample for the case where Q is not a coherentquanti®er. Let A � f1=x1; 1=x2; 0:2=x3g and Q�0� � 0;Q�1=3� � 0:8;Q�2=3� �0:5;Q�1� � 0:2. Clearly, Q is not coherent. Then Z�A� � f0=0; 0=1; 1=2; 0:2=3g.It is easy to check that ZSQ�A� � 0:5 while SQ�A� � 0:8.

5.4. Some examples of application of the discussed methods

We shall use the following ®ve quanti®ers (Fig. 1), that can be de®ned asfollows:

Fig. 1. Five relative quanti®ers.


All�x� � 0; x < 1;1; x � 1;

�Exists�x� � 0; x � 0;

1; x > 0;

�Most�x� � x:

Half�x� � 2� x; x < 0:5;2� �1ÿ x�; x P 0:5;

�At Least Half�x� � 2� x; x < 0:5;

1; x P 0:5:

�The quanti®er Half must be interpreted as ``approximately half'', and is notcoherent. The remaining four quanti®ers are coherent. Because of this, meth-ods that work with coherent quanti®ers will not be used for the quanti®er Half.

Example 5.4.1. Let A be the fuzzy set of Fig. 2. The results obtained using someof the methods and the ®ve quanti®ers de®ned before are shown in Table 1.

A � f0:8=x1; 0:66=x2; 0:58=x3; 0:54=x4; 0:43=x5; 0:4=x6g:We can see that ZQ is a strict method for the quanti®ers Exists and All. Thevalue for SQ is calculated for the quanti®er Half although this method is notde®ned for working with non-coherent quanti®ers. Method YQ cannot beevaluated for the quanti®er Half. Moreover, we can see that forQ �Most; ZQ�A� � YQ�A� � GDQ�A�. We can show that this equality alwaysholds.

Fig. 2. Fuzzy set A for Example 5.4.1.

Table 1

Evaluation of Example 5.4.1

Method Quanti®er

All Exists Most Half At Least Half

ZQ(A) 0 1 0.5683 0.863 1

YQ�A� � CQ�A� 0.4 0.8 0.5683 ± 0.68

GDQ(A) 0.4 0.8 0.5683 0.223 0.68

Ym0Q�A� � SQ�A� 0.4 0.8 0.54 0.66 0.66

ZSQ(A) 0.4 0.8 0.54 0.66 0.66


Proof. First, we have shown that if Q is coherent, YQ�A� � GDQ�A�. As ``Most''is coherent, this part is done. Secondly, let Q �Most; i:e: Q�x� � x. Then wehave ZQ�A� � �P �A�=n� � P �A�=n and

YQ�A� �X

biwi

and

wi � Q�i=n� ÿ Q��iÿ 1�=n� � �i=n� ÿ ��iÿ 1�=n� � 1=n8i;so

YQ�A� � �1=n�X

bi � P �A�=n � ZQ�A�: �

Example 5.4.2. Let A be the fuzzy set de®ned in Fig. 3. The results obtainedusing some of the methods and the ®ve quanti®ers de®ned before are shown inTable 2.

A � f1=x1; 1=x2; 1=x3; 1=x4; 1=x5; 0:9=x6g:We have calculated again the value for SQ(A) with Q�Half and in this ex-ample we can see that this method is not appropriate for evaluating non-coherent quanti®ers like Half, because it is obvious that there are more thanthree elements (the half) in A, so the evaluation cannot be 1.


Table 2


Method Quanti®er


ZQ(A) 0 1 0.983 0.033 1

YQ�A� � CQ�A� 0.9 1 0.983 ± 1

GDQ(A) 0.9 1 0.983 0.033 1

Ym0Q�A� � SQ�A� 0.9 1 0.9 1 1

ZSQ(A) 0.9 1 0.9 0.33 1


Example 5.4.3. Let A be the fuzzy set de®ned in Fig. 4. The results obtainedusing some of the methods and the ®ve quanti®ers de®ned before are shown inTable 3.

A � f0:5=x1; 0:5=x2; 0:5=x3; 0:5=x4; 0:5=x5; 0:5=x6g:

This example shows that ZQ can o�er a strict evaluation even with quanti®ersdistinct of All and Exists. In this case, for the quanti®er Half we haveP �A�=n � 3=6 � 0:5 and Half�0:5� � 1, but it seems to be clear that the fuzzyset A can only have 0 or 6 elements, because if we consider that for examplex1 2 A, then x2 2 A because A�x1� � A�x2�, x3 2 A because A�x1� � A�x3�, etc.and the same consideration holds if x1 62A, so we cannot say that there areexactly 3 elements of A. One interpretation of this behaviour of ZQ can be thatthe cardinality P(A) has problems when adding several little values, because itobtains a value of cardinality that can be unrealistic (in this case, P �A� � 3).These problems can a�ect the behaviour of ZQ. As the fuzzy cardinalities EDand ES used by GDQ and ZSQ only consider 0 and 6 as possible cardinalities ofA and Q�0� � Q�6� � 0 when Q�Half, then the evaluation obtained is 0, andthat seems to be reasonable. For more discussion about the properties of thediscussed cardinalities see [6].


Table 3Evaluation of Example 5.4.3

Method Quanti®er


ZQ(A) 0 1 0.5 1 1

YQ�A� � CQ�A� 0.5 0.5 0.5 ± 0.5

GDQ(A) 0.5 0.5 0.5 0 0.5

Ym0Q�A� � SQ�A� 0.5 0.5 0.5 0.5 0.5

ZSQ(A) 0.5 0.5 0.5 0 0.5


We can also see that ZQ is strict for the coherent quanti®er ``At least thehalf'' in this example, due to the same reasons as in the case of ``Half''. We cansee again that SQ is not appropriate when using Q�Half.

6. New methods for the evaluation of type II sentences

6.1. Generalization of the method GD to type II sentences

A ®rst attempt to generalize the method GD to type II sentences was pro-posed in [4] and it was called G. The aim of this method is to provide a non-strict method (Property 2.2.8) that ful®lls Properties 2.2.2 (when D � X , wehave the method GD for type I sentences), 2.2.3 �O�n logn��, and 2.2.6 (co-herent with logic). First, we de®ne the fuzzy set f �A=D;Q� as

lf �A=D;Q��xi� � oQ�D�xi� ^ A�xi�� 1ÿ oQ��D�xi� ! A�xi��and the evaluation of the sentence Q of D are A is obtained from the evaluationof the sentence ``Q of X are f �A=D;Q�'' using the method GD of Section 4.2, i.e.

GQ�A=D� � GDQ�f �A=D;Q��:However, this method does not ful®ll all the properties of Section 2.2. More-over, this method was not based on the cardinality approach.

The generalized method GD is de®ned as the compatibility between thefuzzy relative cardinality ER and the quanti®er Q by means of the product andthe Lukasiewicz's t-conorm as follows:

GDQ�A=D� �X

c2CR�A=D�ER�A=D; c� � Q�c�:

Property 6.1.1. The method GD verifies Property 2.2.1.

Proof. Let A and D be two crisp sets. Then, by de®nition M(A/D)� {1} som� 1 and a1 � 1 and a2 � 0 and then CR�A=D� � fC�A=D; 1�g i.e.

CR�A=D� � A \ Dj jDj j

� �:

Therefore, ER�A=D; c� � 0 8c 6� C�A=D; 1� and

ER�A=D� � 1jA \ DjjDj

�� and ®nally

GDQ�A=D� � QA \ Dj j

Dj j� �

: �


Property 6.1.2. The method GD for the evaluation of type II sentences in the caseD � X is the method GD for the evaluation of type I sentences described inSection 5.2, and hence fulfills Property 2.2.2.

Proof. We have proved in [6] that if D � X then

ER A=X ;kn

� �� ED�A; k�:

Then we have

GDQ�A=X � �X

c2CR�A=D�ER�A=X ; c� � Q�c�

�X

k2f0;...;ngER�A=X ; k=n� � Q�k=n�

fbecause jX j � n and if c � k=n 62 CR�A=X �then ER�A=X ; c� � 0g

�X

k2f0;...;ngED�A; k� � Q�k=n� � GDQ�A�: �

Property 6.1.3. The method GD has the equivalent expression

GDQ�A=D� �X

ai2M�A=D��ai ÿ ai�1� � Q�C�A=D; ai�

so the efficiency of the method is O�n� if A and D are arranged in decreasingorder, and O�n logn� otherwise. This equivalent expression avoids calculatingER (A/D) explicitly; moreover, GD fulfills Property 2.2.3.

Proof.

GDQ�A=D� �X

c2CR�A=D�ER�A=D; c� � Q�c�

�X

c2CR�A=D�

Xc�C�A=D;ai�

�ai

ÿ ai�1�

!� Q�c�

�X

c2CR�A=D�

Xc�C�A=D;ai�

�ai ÿ ai�1� � Q�C�A=D; ai��


� fby definition of CR �A=D�;every c has associated at least one aig

�X

ai2M�A=D��ai ÿ ai�1� � Q�C�A=D; ai��: �

Property 6.1.4. The method GD fulfills Property 2.2.4.

Proof. Let D Í A. Then A \ D � D and hence M�A=D� � M�D� and 12M(D)(D is normalized). Moreover, CR�A=D� � f1g and hence ER�A=D� � f1=1g, so®nally GDQ�A=D� � Q�1�: �


Proof. Let A \ D � ;. Then, M�A=D� � M�D� and 12M(D) (D is normalized).Moreover, CR�A=D� � f0g and hence ER�A=D� � f1=0g, so ®nallyGDQ�A=D� � Q�0�: �

Property 6.1.6. The method GD fulfills Property 2.2.6 for the quantifier $ bymeans of the t-conorm max and the t-norm min, i.e.

GD9�A=D� � maxxi2X

min A�xi�;D�xi�� :

Proof. We shall proceed in two steps:

($.1) Firstly, we will show that

ER�A=D; 0� � 1ÿmaxxi2X


By de®nition of M(A/D), maxxi2X min A�xi�;D�xi�� max�A \ D� 2 M�A=D�.So, 9j 2 f1; . . . ;mg such that max�A \ D� � aj.

Let i < j. Then, �A \ D� � ;, so A \ D� �aj

�� 0 and hence C�A=D; aj� � 0.

Let i P j. Then �A \ D� 6� ; and hence C�A=D; aj� > 0. Then

ER�A=D; 0� �X

C�A=D;ai��0

�ai ÿ ai�1� �Xi<j

�ai ÿ ai�1�

� �a1 ÿ a2� � �a2 ÿ a3� � � � � � �ajÿ1 ÿ aj� � �a1 ÿ aj�� 1ÿ aj � 1ÿmax�A \ D�� 1ÿmax

xi2Xmin A�xi�;D�xi�� :


($.2) Secondly, we have

GD9�A=D� �X

c2CR�A=D�ER�A=D; c� � 9�c�

�X

c2CR�A=D�; c>0

ER�A=D; c�

�X

c2CR�A=D�ER�A=D; c�

!ÿ ER�A=D; 0�

� fER is a probabilistic measureg

� 1ÿ 1

�ÿmax

xi2Xmin A�xi�;D�xi��

�� max

xi2Xmin A�xi�;D�xi�� : �

For the quanti®er ", we have

GD8�A=D� �X

C�A=D;ai��1

ai� ÿ ai�1�;

i.e. the probability that the relative cardinality of A with respect to D is 1, andhence the probability that D Í A. At this moment, we conjecture that this valuecan be expressed as Property 2.2.6 requires, we hope to o�er a proof of thisconjecture in a future paper.

Property 6.1.7. The method GD fulfills Property 2.2.7, i.e. any quantifier can beused and there is no conflict with any other property.


Proof. We must ®nd fuzzy sets A and D so that the evaluation of the sentence isnot crisp.1. If Q is not crisp, then two integers exist p < q so that 0 < Q�p=q� < 1. Let

c � p=q. We de®ne A and D as follows: A � f1=x1; . . . ; 1=xp; 0=xp�1;. . . ; 0=xng and D � f1=x1; . . . ; 1=xP ;1=xp�1; . . . ; 1=xq; 0=xq�1; . . . ; 0=xng. ThenM�A=D� � f1g; CR�A=D� � fcg; ER�A=D� � f1=cg and GDQ�A=D�� Q�c� with 0 < Q�c� < 1:

2. If Q is crisp, then let Q�0� � w 2 f0; 1g. As Q 6� ; and Q ¹ [0, 1], then twointegers exist p < q so that Q�p=q� � 1ÿ w (i.e. if Q�0� � 0 thenQ�p=q� � 1 and if Q�0� � 1 then Q�p=q� � 0). Let 0 < a < 1 and letc � p=q. Then, let A and D be the following: A � fa=x1; . . . ; a=xp;0=xp�1; . . . ; 0=xng and D � f1=x1; . . . ; 1=xp; 1=xp�1; . . . ; 1=xq; 0=xq�1; . . . ;


0=xng. Then, M�A=D� � f1; ag, and CR�A=D� � f0; p=qg, and ®nallyER�A=D� � f�1ÿ a�=0; a=cg, so ®nally GDQ�A=D� � w� �1ÿ a� ��1ÿ w��a, i.e. if w � 0 then GDQ�A=D� � a, and if w � 1 then GDQ�A=D� �1ÿ a , with 0 < a < 1 and 0 < �1ÿ a� < 1: �

6.2. A possibilistic method for the evaluation of type II sentences

This method was described in [5]. It is based on the cardinality approach,and uses the relative cardinality ES described in Section 4.2.2. The resultingmethod, also called ZS, is de®ned as follows

ZSQ�A=D� � maxc2CR�A=D�

min ES�A=D; c�;Q�c��

i.e. the cardinality approach using max±min composition to obtain the com-patibility between the (fuzzy) relative cardinality and the quanti®er.

Property 6.2.1. The method ZS verifies Property 2.2.1.

Proof. Let A and D be two crisp sets. Then, by de®nition M(A/D)� {1} andthen

CR�A=D� � A \ Dj jDj j

� �

therefore

ES�A=D� � 1jA \ DjjDj

�� and ®nally

ZSQ�A=D� � QA \ Dj j

Dj j� �

:

Property 6.2.2. The method ZS for the evaluation of type II sentences in the caseD � X is the method ZS for the evaluation of type I sentences described inSection 5.3, and hence fulfills Property 2.2.2.

Proof. We have proved in [6] that if D � X then

ES A=X ;kn

� �� Z�A; k�:


Let S(A) be the support of A. Then

ZSQ�A=X � � maxc2CR�A=X �

min ES�A=X ; c�;Q�c��

� maxk2S�A�

min ES A=X ;kn

� �;Q�k=n�

� �� max

k2S�A�min Z�A; k�;Q�k=n��

� maxk2 0;...;nf g

min Z�A; k�;Q�k=n�� ZSQ�A�: �

Property 6.2.3. The method ZS has the equivalent expression

ZSQ�A=D� � maxa2M�A=D�

min a;Q�A \ D�a��

Daj j� ��

so the efficiency of the method is O�n� if A and D are arranged in decreasingorder, and O�n logn� otherwise. This equivalent expression avoids calculatingES(A/D) explicitly; moreover, ZS fulfills Property 2.2.3.

Proof. Analogous to Property 5.3.4.

ZSQ�A=D� � maxc2CR�A=D�

min ES�A=D; c�;Q�c��

� maxc2CR�A=D�

min max a 2 M�A=D� c

�� A \ D� �a��

Daj j�; Q�c�

�� max

c2CR�A=D�max

c� A\D� �aj jDaj j

min a;Q�c��

� maxa2M�A=D�

min a;QA \ D� �a��

Daj j� ��

: �


Proof. Let D Í A. Then A \ D � D and hence M�A=D� � M�D� and 1 2 M�D�(D is normalized). Moreover, CR�A=D� � f1g and hence ES�A=D� � f1=1g, so®nally ZSQ�A=D� � Q�1�: �


Proof. Let A \ D � ;. Then, M�A=D� � M�D� and 1 2 M�D� (D is normalized).Moreover, CR�A=D� � f0g and hence ES�A=D� � f1=0g, so ®nallyZSQ�A=D� � Q�0�: �Property 6.2.6. The method ZS fulfills Property 2.2.6 for the quantifiers $ and ".In the first case, by means of the t-conorm max and the t-norm min, i.e.


ZS9�A=D� � maxxi2X


In the second case, by means of the t-norm min and a family of implications

ZS8�A=D� � a � minxi2X

Ia D�xi�;A�xi�� ;

where Ia�d; a� is defined by

Ia�d; a� �1; d 6 max�a; a�; d < 1;a; a6 a < d < 1;a otherwise:

8<:We will show that Ia is a fuzzy implication for all a2 [0, 1] in Appendix A.

Proof.

�9� ZS9�A=D� � maxa2M�A=D�

min a ; 9 �A \ D�a��

Daj j� ��

� max a 2 M�A=D� �A \ D�a 6� ;��

� maxxi2X

�A \ D��xi�f g� max

xi2Xmin A�xi�;D�xi�� f g:

�8� Let X 1 � x 2 X j A�x� < D�x�f g and let X 2 � X n X 1: We will make theproof in four steps :�8:1� Firstly; we will show that 9x0 2 X so that ZS8�A=D� � A�x0�:

· Let ZS8�A=D� � 1: Then A1 � D1 6� ; and 9x0 2 X so that A�x0� � D�x0� �1 � ZS8�A=D�:

· Let ZS8�A=D� < 1: Then X 1 6� ; �if X 1 � ; then D � A and by Property6:2:4; ZS8�A=D� � 8�1� � 1� and

ZS8�A=D� � maxa2M�A=D�

min a; 8 A \ D� �a��

Daj j� ��

� max a 2 M�A=D�j �A \ D�a � Da

� � max

x2Xa 2 a�x�; d�x�f g such that �A \ D�a � Da

� � max

x2Xa 2 a�x�; d�x�f g such that �AjX 1 \ DjX 1�a [ �AjX 2 \ DjX 2�a�

� �DjX 1�a [ �DjX 2�ag:

By de®nition of X 2; AjX 2 \ DjX 2 � DjX 2 so we can say ZS8�A=D� �maxx2X 1fa 2 fa�x�; d�x�g such that �AjX 1 \ DjX 1�a � �DjX 1�ag. We knowA�x� < D�x�8x 2 X 1; so ZS8�A=D� � maxx2X 1fa � a�x� such that �AjX 1 \ DjX 1�a� �DjX 1�ag and x0 � maxx2X 1fa�x�j �AjX 1\DjX 1�a � �DjX 1�ag:


�8:2� Secondly, we will show that if ZS8�A=D� � a thenminx2XfIa�D�x�;A�x��gP a.

Let us suppose minx2XfIa�D�x�;A�x��g � A�x1� < a: Then by de®nition ofIa; D�x1� � 1: But then x1 2 D1 and hence x1 2 Da: By de®nition ofa; Aa \ Da � Da; so x1 2 Aa; and hence A�x1�P a (contradiction), so we con-clude minx2XfIa�D�x�;A�x��gP a:�8:3� Thirdly, we will show that 9x0 2 X so that if ZS8�A=D� � a then

Ia�D�x0�;A�x0�� a:· Let a � 1: Then by �8:1� 9x0 2 X so that A�x0� � D�x0� � 1: Then,

Ia�D�x0�;A�x0�� I1�1; 1� � 1 � a:· Let a < 1: Then by �8:1� 9x0 2 X 1 � X so that A�x0� � a: As

x0 2 X 1; A�x0� < D�x0�; so we have A�x0� � a < D�x0�:1. Let D�x0� � 1: Then, Ia�D�x0�;A�x0�� Ia�1; a� � a:2. Let D�x0� < 1: Then we have A�x0� � a < D�x0� < 1; so by de®nitionIa�D�x0�;A�x0�� a:

�8:4� Finally, we have shown in �8:2� that minx2XfIa�D�x�;A�x��gP a and in�8:3� that 9x0 2 X so that if ZS8�A=D� � a then Ia�D�x0�;A�x0�� a; so we haveZS8�A=D� � a � minxi2X Ia�D�xi�;A�xi��: �

Property 6.2.7. The method ZS fulfills Property 2.2.7, i.e. any quantifier can beused and there is no conflict with any other property, except in the case of a crispquantifier verifying Q�0� � Q�1� � 1, as will be discussed in Property 6.2.8.

Property 6.2.8. The method ZS fulfills Property 2.2.8 except in the case that Q isa crisp quantifier and Q�0� � Q�1� � 1.

Proof. We must ®nd fuzzy sets A and D so that the evaluation of the sentence isnot crisp.1. If Q is not crisp, then two integers exist p < q so that 0 < Q�p=q� < 1. Let

c � p=q. We de®ne A and D as follows: A � f1=x1; . . . ;1=xp;0=xp�1; . . . ; 0=xng and D � f1=x1; . . . ; 1=xp; 1=xp�1; . . . ; 1=xq;0=xq�1; . . . ;0=xng. Then M�A=D� � f1g, CR�A=D� � fcg, ES�A=D� � f1=cg andZSQ�A=D� � Q�c�:

2. If Q is crisp and Q�0� < 1, then we consider two cases:(a) Q�0� > 0. Then we de®ne A � ; and D � X � f1=x1; . . . ; 1=xng. HenceM�A=D� � f1g; CR�A=D� � f0g; ES�A=D� � f1=0g and ZSQ�A=D� � Q�0�:(b) Q�0� � 0. By hypothesis of Property 2.2.8, Q 6� ; and hence twointegers exist p < q so that Q�p=q� � 1 (Q is crisp). Let c � p=qand let 0 < a < 1. We de®ne A and D as follows: A � fa=x1; . . . ; a=xp;0=x� p � 1; . . . ; 0=xng and D � f1=x1; a=x2; . . . ; a=xp; a=xp�1; . . . ; a=xq;0=xq�1;. . . ; 0=xng. Then M�A=D� � f1; ag; CR�A=D� � f0; cg;ES�A=D� � f1=0; a=cgand ZSQ�A=D� � maxfmin�1;Q�0��; min�a;Q�c��g � min�a;Q�c�� a:


3. If Q is crisp and Q�1� < 1 then we consider two cases:(a) Q�1� > 0. Then we de®ne A � D � Xf1=x1; . . . ; 1=xng. ThenM�A=D� � f1g; CR�A=D� � f1g; ES�A=D� � f1=1g and ZSQ�A=D� � Q�1�:(b) Q�1� � 0. By hypothesis of Property 2.2.8, Q 6� ; and hence twointegers exist p < q so that Q�p=q� � 1 (Q is crisp). Let c � p=q and let0 < a < 1. We de®ne A and D as follows: A � f1=x1; a=x2 . . . ; a=xp;0=xp�1

; . . . ; 0=xng and D � f1=x1; a=x2; . . . ; a=xp;a=xp�1; . . . ; a=xq;0=xq�1; . . . ; 0=xng:Then M�A=D� � f1; ag; CR�A=D� � f1; cg; ES�A=D� � f1=1; a=cg andZSQ�A=D� � maxfmin�1;Q�1��; min�a;Q�c��g � min�a;Q�c�� a: �

The question is, what could be the semantic of a crisp quanti®er havingQ�0� � Q�1� � 1? We think that this strange case will not be used in practice,so that this ``exception'' does not a�ect the ful®llment of Property 6.2.8 by themethod ZS.

6.3. Some examples of application of the discussed methods

We shall use the quanti®ers of Fig. 1, as in Section 5.4.

Example 6.3.1. Let A and D be the fuzzy sets de®ned in Fig. 5. The resultsobtained using some of the methods and the ®ve quanti®ers de®ned before areshown in Table 4.

A � f1=x1; 1=x2; 1=x3; 1=x4; 1=x5; 0:9=x6gD � f0:3=x1; 0:4=x2; 0:8=x3; 1=x4; 0:1=x5; 0:2=x6g:

In this example, D Í A and D is normalized, so for every quanti®er Q the ex-pected result is Q (1), i.e. All (1)� 1, Exists (1)� 1, Most (1)� 1, Half (1)� 0and At Least Half (1)� 1. We can see that only ZQ, GDQ and ZSQ verify thisproperty in general. The rest of the methods fail in this example for quanti®ersAll and Most.

Fig. 5. Fuzzy sets A and D of Example 6.3.1.


Example 6.3.2. Let A and D be the fuzzy sets de®ned in Fig. 6. The resultsobtained using some of the methods and the ®ve quanti®ers de®ned before areshown in Table 5.

A � f0=x1; 0=x2; 0=x3; 1=x4; 1=x5; 1=x6g;D � f1=x1; 1=x2; 0:1=x3; 0=x4; 0=x5; 0=x6g:

In this example, we can see that method YQ does not ful®ll Properties 2.2.6-$and 2.2.5, because as A \ D � ; then the evaluation must be Q (0), and we haveAll (0)�Exists (0)�Most (0)�Half (0)�At Least Half (0)� 0.


Table 4


Method Quanti®er


ZQ(A/D) 1 1 1 0 1

YQ(A/D) 0.9 1 0.964 ± 1

VQ(A/D) 0.9 1 0.95 ± 0.98

GDQ(A/D) 1 1 1 0 1

ZSQ(A/D) 1 1 1 0 1

Table 5


Method Quanti®er


ZQ(A/D) 0 0 0 0 0

YQ(A/D) 0 0.9 0.042 ± 0.085

VQ(A/D) 0 0 0 ± 0

GDQ(A/D) 0 0 0 0 0

ZSQ(A/D) 0 0 0 0 0


Example 6.3.3. Let A and D be the fuzzy sets de®ned in Fig. 7. This examplehas been extracted from [21]. The results obtained using some of the methodsand the ®ve quanti®ers de®ned before are shown in Table 6.

A � f0:8=x1; 0:4=x2; 0:9=x3; 1=x4; 1=x5g;D � f0:6=x1; 0:3=x2; 1=x3; 0=x4; 0:1=x5g:

In this example we can see that ZQ is too strict with the quanti®er All.

7. Conclusions and future works

In this paper, we propose a (not closed) set of appropriate properties to beful®lled by any method of evaluation of type I and type II sentences. We havediscussed some existing methods from this point of view. For type I sentences,some of the existing methods are satisfactory with respect of most of theproperties related to the evaluation, although they are restricted to relative andcoherent quanti®ers. For type II sentences, the existing methods are notsatisfactory in the evaluation, and are also restricted to coherent quanti®ers (aswe discussed in Section 1, type II sentences are evaluated only for relative


Table 6


Method Quanti®er


ZQ(A/D) 0 1 0.95 0.1 1

YQ(A/D) 0.7 1 0.775 ± 0.85

VQ(A/D) 0.7 0.9 0.8 ± 0.86

GDQ(A/D) 0.9 0.9 0.9 0 0.9

ZSQ(A/D) 0.9 0.9 0.9 0 0.9


quanti®ers). We have chosen the cardinality approach to obtain new methodsthat ful®ll all the properties we have considered, and we have also interpretedthese methods in terms of the cardinality approach.

We have shown that our method GD for type I sentences is a generalizationof YagerÕs method based on OWA (that is, the method based on the Choquetfuzzy integral) that allows the use of any quanti®er, wether coherent or not,and that can be used with absolute quanti®ers. We have also shown that ourmethod ZS for type I sentences is a generalization of the method based on theSugeno fuzzy integral that allows using the same quanti®ers as GD. Bothmethods GD and ZS ful®ll Properties 2.1.1±2.1.6 for every absolute and rel-ative quanti®er. These methods use some new de®nitions of fuzzy cardinalityde®ned in [6]. Another contribution has been an interpretation of the methodbased on the Sugeno integral by means of the cardinality approach usingDubois±Prade fuzzy cardinality and max±min composition.

Our contribution for the evaluation of type II sentences are the methods ZSand GD for type II sentences, which are e�cient and non-strict methods ofevaluation, ful®lling Properties 2.2.1±2.2.8 that we propose for every relativequanti®er. They are also based on the cardinality approach, using newde®nitions of fuzzy relative cardinality proposed in [6]. Tables 7 and 8 show themethods discussed and proposed in this paper together with the proper-ties ful®lled by each one. An ``X'' means that the method ful®lls the property.

Type I sentences: As we discussed before, the e�ciency of the methods GDand ZS can be improved to O�n� if the fuzzy set A is arranged in non-increasingorder.

Table 7

Properties of type I sentences evaluation methods

Method 2.1.1 2.1.2 2.1.3 2.1.4 2.1.5 2.1.6

Zadeh X X O�n� X

Yager-OWA X X X O�n log n� X

GD X X X O�n logn� X X

ZS X X X O�n logn� X X

Table 8

Properties of type II sentences evaluation methods

Method 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.2.6.-$ 2.2.6.-" 2.2.7 2.2.8

Zadeh X X O�n� X X X

Yager X X O�n logn� X X

Vila O�n� X X X

GD X X O�n logn� X X X ? X X

ZS X X O�n logn� X X X X X X


Type II sentences: The same consideration can be made with respect to thee�ciency and the ordering of the fuzzy sets A and D. The e�ciency of themethods GD and ZS is O�n� if D and A are arranged in non-increasing order.

We have developed two methods for every type of quanti®ed sentence, oneprobabilistic method (GD) and one possibilistic method (ZS) that generalizethe existing probabilistic (Choquet) and possibilistic (Sugeno) approaches forthe evaluation of type I sentences to type II sentences and any type of quan-ti®er. These methods are based on new de®nitions of fuzzy cardinalities that arealso related in terms of generalization. Schema (Fig. 8) shows the relationbetween the methods and between the cardinalities. The meaning of the arrowsX ® Y is ``X generalizes Y''.

Future works will focus on the e�cient implementation of the new methodsproposed and its use in database tasks and applications such as ¯exible queryand data mining, where some new models which we are developing are basedon the evaluation of quanti®ed sentences. Another future work will be to ®ndthe relation between method GD and Property 2.2.6-".

Appendix A

We will show that the family of functions Ia de®ned in Property 6.2.6 is afamily of fuzzy implications. One fuzzy implication is a function with�0; 1� � �0; 1� ! �0; 1� verifying the following ®ve properties (see [16]):1. Let c6 b. Then, I�c; x�P I�b; x�;2. Let c6 b. Then, I�x; c�6 I�x; b�;3. I�0; x� � 1;

Fig. 8. Relations among methods of evaluation, with their corresponding cardinalities.


4. I�1; x� � x;5. I�b; I�c; x�� I�c; I�b; x��:

The family Ia is de®ned as follows:

Ia�d; a� �1; d 6 max�a; a�; d < 1;a; a6 a < d < 1;a otherwise;

8<:where a2 [0, 1].

Property A.1. Let c6 b. Then, we consider three cases:(a) Let Ia�c; x� � 1. Then it is obvious that I�c; x�P I�b; x�:(b) Let Ia�c; x� � a. Then, x 6 a < c< 1, so x 6 a < b 6 1.

If b� 1 then Ia(b,x)� x 6 a� Ia(c,x).If b < 1 then x 6 a < b < 1 so Ia(b,x)� a� Ia(c,x).

(c) Let Ia�c; x� � x. There are only two possibilities:If c � 1 then b � 1 and then Ia�b; x� � x � Ia�c; x�:If c < 1 and c > a and x > a, then b > a and then Ia�b; x� � x � Ia�c; x�:

Property A.2. Let c6 b. Then, we consider three cases:(a) Let Ia�x; b� � 1. Then it is obvious that I�x; b�P I�x; c�:(b) Let Ia�x; b� � a. Then c 6 b 6 a < x < 1, so Ia�x; b� � a � Ia�x; c�:(c) Let Ia�x; b� � b. Then there are three possibilities:

If x � 1 then Ia�x; c� � c6 b � Ia�x; b�:If x < 1 and x > a and b P c > a then Ia�x; c� � c6 b � Ia�x; b�:If x < 1 and x > a and b P aP c then Ia�x; c� � a6 b � Ia�x; b�:

Property A.3. Ia�0; x� � 1 because 06 max�x; a� and 0 < 1:

Property A.4. Ia�1; x� � x because if d � 1 then Ia�d; a� � a:

Property A.5. We must prove that Ia�b; Ia�c; x�� Ia�c; Ia�b; x��. We shall con-sider six cases:

1. Let Ia�c; x� � Ia�b; x� � 1. Then, Ia�b; Ia�c; x�� Ia�b; 1� � 1 andIa�c; Ia�b; x�� Ia�c; 1� � 1:

2. Let Ia�c; x� � 1 and let Ia�b; x� � x < 1. Then, Ia�b; Ia�c; x�� Ia�b; 1� � 1and Ia�c; Ia�b; x�� Ia�c; x� � 1:

3. Let Ia�c; x� � Ia�b; x� � x < 1. Then Ia�b; Ia�c; x�� Ia�b; x� � x andIa�c; Ia�b; x�� Ia�c; x� � x:

4. Let Ia�c; x� � 1 and let Ia�b; x� � a < 1. Then, Ia�b; Ia�c; x�� Ia�b; 1� � 1.On the other hand, Ia�c; x� � 1 so c6 max�x; a� and c < 1. Moreover,


Ia�b; x� � a so x6 a < b < 1 and hence max�x; a� � a < 1, so we havec6 max�a; a� and c < 1 so Ia�c; Ia�b; x�� Ia�c; a� � 1:

5. Let Ia�c; x� � Ia�b; x� � a < 1. Then x 6 a < b < 1 and x 6 a < c < 1 soa � a < b < 1 and a � a < c < 1 and hence Ia�c; Ia�b; x�� Ia�c; a� � aand Ia�b; Ia�c; x�� Ia�b; a� � a:

6. Let Ia�c; x� � x < 1 and let Ia�b; x� � a < 1. Then x 6 a < b < 1 andIa�b; Ia�c; x�� Ia�b; x� � a. On the other hand, as Ia�c; x� � x < 1 then we haveeither c > max�x; a� or c � 1.

Let c � 1. Then Ia�c; Ia�b; x�� Ia�c; a� � Ia�1; a� � a:Let c > max�x; a� and c < 1. Then we have x 6 a < c < 1 (we knewx 6 a < b < 1) and hence Ia�c; x� � a, so x � a and henceIa�c; Ia�b; x�� Ia�c; a� � Ia�c; x� � x � a:

Table 9 is a summary of the six cases discussed before.The rest of the cases are symmetrical with respect to these six cases (inter-

changing b and c).This family of implications also veri®es another property:

Property A.6. Ia�x; x� � 1 (identity principle). We shall discuss two cases:Let x < 1. Then x6 max�x; a� and x < 1 so Ia�x; x� � 1:Let x � 1. Then by Property A.4, Ia�1; 1� � 1:

Special cases are as follows:1. If a � 0 then we obtain

I0�d; a� � 1; d 6 a;a; d > a;

�

which is called a G�odel implication. This is an R-implication. An R-implicationcan be de®ned by means of the expression

Table 9

Possible cases for Property A.5

Ia�c; x� Ia�b; x� Ia�b; Ia�c; x�� Ia�c; Ia�b; x��1 1 1 1

1 x 1 1

x x x x

1 a 1 1

a a a ax a a a


I�d; a� � sup xjt�d; x�f 6 ag;

t being a continuous t-norm. For this G�odel implication, t is the minimum.Since the greatest t-norm is the minimum, this implication is the greatest lowerbound of R-implications.

2. If a � 1, then we obtain the implication

I1�d; a� � 1; d < 1;a; d � 1;

�

which is considered as the least upper bound of the class of R-implications (see[20]) although it cannot be de®ned using a continuous t-norm by means of theexpression de®ned before. Nevertheless, a non-continuous t-norm exists so thatI1 can be de®ned by means of the expression for R-implications. This is thedrastic intersection

t�x; y� �x; y � 1;

y; x � 1;

0 otherwise:

8><>:In fact, it can be shown that any implication of the family Ia can be de®ned bymeans of the expression for R-implications using the following family of (non-continuous in general) t-norms:

ta�x; y� �y; x � 1;

x; y � 1;

min�x; y�; a < x; y < 1;

0 otherwise;

8>><>>:so that Ia�d; a� � supfxjta�d; x�6 ag.

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Fuzzy cardinality based evaluation of quantified sentences

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